The present invention relates to a method for improving reconstructed NMR images and, more specifically, to a method for cancelling ghosts from NMR images.
Under ideal conditions, reconstructed NMR images must be positive and real. This is because, in NMR experiments, the observed signal is Fourier transform of density distribution of the object under consideration, which by definition is a real positive quantity. In practice however, even the most straightforward ways of scanning the k.sub.x -k.sub.y space, (e.g., row by row or column by column scanning without date reversal), result in complex images. This is partly due to a shift in the data from the notional origin in k space. If readout gradient is constant and uniform time samples of data are inverse Fourier transformed to generate the image, delay in the time data translates into a linear phase shift in the reconstructed image. These phase shifts can be easily determined and eliminated, since they do not affect the magnitude of the reconstructed images.
When NMR data is obtained by scanning the k.sub.x -k.sub.y space, with data reversal on alternate y lines (where y is the horizontal axis across which readout occurs), time delays between the start of data acquisition and the start of the readout pulse are different for even and odd lines. The effect of this on the image manifests itself as a ghost separated by half the image size. Under these conditions, if readout gradient is constant and data is sampled uniformly in time, then the ghost image can be entirely removed by a first-order phase difference between odd and even lines. However, when the readout gradient is sinusoidal and the image is reconstructed by inverse Fourier transforming non-uniform samples (see, e.g., the method disclosed in U.S. Pat. No. 4,740,748) the difference between even and odd line delays degrades introduces ghosts and thus the quality of the resolution.
This, together with asymmetry of the sinusoidal readout gradient for even and odd lines can be modeled by multiplying even and odd parts of the NMR image by two separate phase functions .phi. (n.sub.1,n.sub.2) and .theta. (n.sub.1,n.sub.2). More specifically, if x(n.sub.1,n.sub.2) denotes the true density distribution of the object under consideration, and Y(n.sub.1,n.sub.2) denotes the 2-D inverse discrete Fourier transform of the time data (i.e., the reconstructed ghosted image), then the even and odd parts of the observed image, Y.sub.even (n.sub.1,n.sub.2) and Y.sub.odd (n.sub.1,n.sub.2) can be modeled as: ##EQU1## where the dimensions of the reconstructed image are N.times.N.sub.s, and the number of echoes is N. (Note that if the sinusoidal readout y gradient was identical for even and odd lines, then the even and odd phase functions would have been identical. That is, .phi. (n.sub.1,n.sub.2).ident..theta. (n.sub.1,n.sub.2)).
Having modeled the ghosted image, the objective can be stated as estimation of the true object density distribution x(n.sub.1,n.sub.2) from the observed ghosted image Y(n.sub.1,n.sub.2).
From equations (1) and (2), it is clear that if the phase functions .phi. (n.sub.1,n.sub.2) and .theta.(n.sub.1,n.sub.2) are known for all values of n.sub.1 and n.sub.2, then x(n.sub.1,n.sub.2) and x(n.sub.1,n.sub.2 +N/2) can be determined from Y.sub.even (n.sub.1,n.sub.2) Y.sub.odd (n.sub.1,n.sub.2) by solving a 2.times.2 linear system of equations. In practice, the difference between .phi. (n.sub.1,n.sub.2) and .theta. (n.sub.1,n.sub.2) can be determined experimentally by placing a test object in the upper and lower half of the field of view (FOV) and measuring the difference between even and odd parts of the resulting images. More specifically, when the object is in the upper half of the FOV, by definition: EQU x(n.sub.1,n.sub.2 +N/2)=0
Substituting this into equations (1) and (2): ##EQU2##
The phase difference between Y.sub.even (n.sub.1,n.sub.2) Y.sub.odd (n.sub.1,n.sub.2) can be used to obtain ##EQU3##
Similarly, by placing the object in the lower half of the FOV: EQU x(n.sub.1,n.sub.2)=0 ##EQU4##
The phase difference between Y.sub.even (n.sub.1,n.sub.2) and Y.sub.odd (n.sub.1,n.sub.2) can be used to obtain ##EQU5##
Thus, experimental values of .DELTA. (n.sub.1,n.sub.2) and .DELTA. (n.sub.1,n.sub.2 +N/2) can be used to determine A(n.sub.1,n.sub.2) and B(n.sub.1,n.sub.2) by solving the above linear system of equations. Once A and B are determined, their magnitudes can be used to find x(n.sub.1,n.sub.2) and (n.sub.1,n.sub.2 +N/2) respectively.
Experimentally, there are two major drawbacks with the above approach. The first drawback has to do with the fact that the phase difference function is a function of the parameters for the NMR experiments. Some of these parameters are the strength of the x, y and z gradients, and the static magnetic field or the RF. Therefore, to be able to apply this method successfully, a different look up table is needed for different experimental set ups. The second drawback has to do with the fact that the phase difference function .DELTA. (n.sub.1,n.sub.2) is somewhat object dependent. More specifically, although the general shape of .DELTA. (n.sub.1,n.sub.2) does not vary drastically from one object to the next, the change is large enough to introduce considerable amount of ghosts. The third drawback of the above approach has to do with the fact that obtaining the phase difference function .DELTA. (n.sub.1,n.sub.2) of a test object for all values of n.sub.1 and n.sub.2 is a non-trivial task from an experimental point of view. This has to do with factors such as susceptibility effects.
In short, it has been found that the performance of the above scheme is inadequate for most ghosted images. Accordingly, it would be highly desirable to process NMR signals using a method of ghost correction in the form of an algorithm which is automatic and does not require a look up table.